On the equivalence of the Impulse Approximation and the Gersch-Rodriguez-Smith theory for structure functions

We derive for a non-relativistic system an approximation for Final State Interactions in a form, resembling a DWIA which corrects the structure function computed in the PWIA. We then compare the Gersch-Rodriguez-Smith and the IA series for structure functions and prove that to order ${\cal O}(1/q^2)$ the above DWIA representation is contained in the GRS series to the same order. There is an additional term in the GRS series that is missing in the DWIA due to the eikonal approximations in the latter. This strongly suggests that the two approaches, when treated exactly, produce identical structure function to arbitrary order in $1/q$.